Let \(\beta >0\) , and, for all \(x\ge 0\) , let \(\psi _H(x)=x\,\mathbb {1}\lbrace x\le \beta \rbrace +\beta \,\mathbb {1}\lbrace x>\beta \rbrace \) .
From Lemma REF , Assumptions REF  hold in this example with \(\gamma =1\) .
As \(\psi _H\)  is bounded by \(\beta \) , Hoeffding's lemma (see [1]}) shows that \(\max _{1\le i\le n}\Vert \psi _H(\Vert X_i-T_H(P)\Vert )\Vert _{\psi _1}\le \beta \) .
